ICTP/SISSA Joint Colloquium in Mathematics on 12 December

[Math]

ICTP/SISSA Joint Colloquium in Mathematics


Wednesday, 12 December 2007, at 15.00 hrs.


Professor Vladimir I. Arnold
Steklov Mathematical Institute, Moscow, Russia


"Measurements of randomness"

Summary: The objective decision whether a given sequence consists of  
random or of non-random numbers is not easy, but Kolmogorov published  
(in Italian) a paper on this problem in an insurance statistics  
journal (G. ist. ital. attar. 1933).

The ramdomness measure introduced in that paper suggests that the  
geometric progressions like

03, 09, 27, 81, 43, 29, 87, 61, 83, 49, 47, 41, 23, 69, 07

are more random than the arithmetical ones like

37, 74, 11, 48, 85, 22, 59, 96, 33, 70, 07, 44, 81, 18, 55.

For the arithmetical progressions of fractional parts, whose  
difference is a rational number, the ramdomness parameter tends to  
zero when the length of the progression grows.

There exist irrational differences for which the randomness  
parameters of long arithmetical progressions of fractional parts do  
not tend to zero and whose behaviour is unknown for almost all values  
of the difference because of some unsolved problems of the statistics  
of continued fractions.

The most known use of the Kolmogorov Italian paper to practical  
problems was his 1940 study of the attempts by Lyzenko to reject the  
Mendel law of genetics, which was based on the empirically observed  
difference between the experimental data and the prediction of  
Mendel's law.

Kolmogorov proved that the experiences of the Lyzenko school  
confirmed the Mendel law rather than reject it: a smaller difference  
between the theory and the experiment would be a proof of the  
falsification of the experimental data.

These conclusions of Kolmogorov were, however, unpublished, since the  
classical genetic experts answered to Lyzenko by their own  
experimental data, whose difference with Mendel's prediction were  
minimal.


Venue:  ICTP Main Lecture Hall, Main Building, ground floor